IG 1628 
G62 
.887 
lopy 1 



LIBRARY OF CONGRESS 



021 060 446 7 






REVISED AND ENLARGED EDITION. 

SHORT METHODS 

FOR 

COMPUTING 

INTEREST rf" DISCOUNT. 

Simple Interest, Discount, Compound Interest, 



^ 



HENRY GOLDMAN, 

AUTHOR OF 

"The Arithmetical Detector," 
" New Method for Averaging Accounts, '' 
•Combination Discount Calculator," "The Expert Calculato 
Etc., Etc. 






, AUG 1 1887 <P; 

CHICAGO. 1887. 






.£4>Z 
IS*7 



COPYRIGHT, 1884, BY HENRY GOLDMAN. 
ALL RIGHTS RESERVED. 



LC Control Number 




tmp96 027241 



Address all orders, communications or inquiries, to 

HENRY GOLDMAN, Author and Publisher, 

J 16 LaSalle Street, Chicago, II 



P"W£. 



A CCURACY and speed in computing interest and dis- 
-^—*- count are important accomplishments; that they are 
so rarely found, proves to a great extent the deficiency of the 
methods hitherto introduced. 

The author offers in the following pages a method, pro- 
nounced by experts as the shortest and simplest interest method 
known, and which, according to his own judgment, leaves 
nothing to be desired. It can be easily acquired, and when 
once learned, is hardly ever to be forgotten. Being applica- 
ble to all cases which may possibly arise, it is bound to win its 
way into general use. 

The only objection against the "ioo Days Interest Method," 
that it starts with a division, while the 60 days rule presents an 
immediate basis, is very shallow, and only illustrates the 
poor judgment of those who most persistently urge it. In the 
first place, has the initial division a tendency to reduce the 
whole calculation. Secondly, does the decimal basis arrived 
at decrease all mental labor to a possible minimum. And 
last, but not least, is the one division in the beginning a pre- 
ventive for the many divisions, which form so great a detriment 
to the 60 days rule, and, through the repeated loss of frac- 
tions, make it for that rule almost impossible to give an abso- 



lutely or even practically correct answer. These are con- 
clusive arguments which cannot be effectively contradicted. 

Claiming the universal application of the ioo days basis for 
computing interest as my own original idea, I do not arrogate 
to myself the first knowledge of the fact that any principal 
divided by 6 gives the interest at 6 percent, for 1,000 days. I 
simply contend that no work published previous to the date 
of my copyright (February 27th, 1884) contains a method 
which makes an equally uniform and exclusive use of 100 
days as a starting point, though since the publication of this, 
?7iy " 100 Days Interest Method," and its introduction to 
thousands of book-keepers throughout the United States and 
Canada, some irresponsible parties have, for obvious reasons, 
shamelessly copied it. I cannot reach those sneaking imitat- 
ors, but will prosecute all financially responsible infringers to 
the full extent of the law. 

For bank discount no more satisfactory method has ever 
been invented. Based on a strictly scientific principle, it com- 
bines all the advantages of the so-called table methods without 
sharing any of their faults. The time required for opening a 
table, looking for page, etc., is more than sufficient to ascer- 
tain the result by this new method, saving time and inconven- 
ience. 

As another advantage, the author claims that errors result- 
ing from the neglect of fractions, which, in the aggregate of 
many items, may cause a considerable difference, are practically 
avoided by a characteristic reduction, decreasing all variations 
from three to twelve times. The consideration of the tens of 
the cents of the principal, whenever the number of days ex- 
ceeds 100, is another of the many good features of this meth- 



od, while the exclusive application of decimals places it 
at the head of all methods previously taught. 

A simplified rule for ascertaining the present value of a note 
and an improved method for compound interest rank also 
among the most original parts of this work. 

The striking advantages derived from adopting the methods 
herein given will convince even the most skeptical of their 
excellence. 

The Author. 



C0fi5yl5S. 



Page 

Introduction, 7 

To Compute the Interest. 

The Time expressed in Days, 9 

The Time expressed in Months, 14 

The Time expressed in Years, 15 

To Find the Principal, 15 

To Find the Rate, 16 

To Find the Time, 16 

Discoi>\ r m. 

Mercantile Discount. 

Simple Discount, 17 

Combination Discount, 18 

Bank Discount. 

The Year figured at 360 days, 19 

The Year figured at 365 days, 20 

True Discount. 

To Ascertain the Amount, 20 

To Ascertain the Principal, 21 

To Find the Compound Interest, 21 

To Find the Principal, 23 

To Find the Rate, 23 

To Find the Time, 24 
Annuities. 

To Find the Amount, 24 

To Find the Annuity, 24 



[J^ODUQJIOli 



Interest is a compensation for the use of money. 
Principal — The money on which interest is com- 
puted. Rate Per Cent. — The number of cents 
paid for the use of Si.oo during one year, if not 
otherwise specified. Amount — The sum of princi- 
pal and interest. 

Accurate Interest being calculated on the ba- 
sis of 365 days to the year, is one seventy-third part 
smaller than Usual Interest, which is computed 
on the basis of 360 days. * 

Interest on principal only is also called Simple 
Interest, to distinguish it from Compound Inter- 
est, which is interest on the sum of a given princi- 
pal and its accumulated interest. 

Discount is an allowance for the payment of 
money before due. Face Value — The amount of 
bill, note, etc., from which discount is deducted. 
Net Proceeds — The difference between face 
value and discount. 

Discount deducted from the face of a bill, in- 
voice, etc., without special reference to time, is 
called Mercantile Discount, while Bank Dis- 
count is a deduction from the face value of a note, 
draft, etc., equivalent to the interest for the exact 
time from date of discount to due date, including 
three days as days of grace allowed by law. 

The principal which, at a given rate and in a given 
time, produces a given amount, represents the Pres- 



ent Value of the amount and leaves, deducted from 
the latter, the True Discount. "Bank Discount," 
and "True Discount" are, therefore, essentially dif- 
ferent. 

The "ioo Days Interest Method," forming the 
most important part of this work, will prove its su- 
periority over the 30 or 60 days rule in offering a 
threefold basis — the interest for 100 days, 10 days 
and 1 day — which facilitates the computation of in- 
terest for any odd number of days and dispenses 
with the awkward subdivision into aliquot parts, re- 
quired by the 6 and 12 per cent, rules. Though in 
a number of special cases the application of the 
above-named rules will recommend itself, the gen- 
eral usefulness and time and labor saving properties 
of the 100 days method have never been approached. 

To the following rates and corresponding numbers 
of days only, the "100 Days Interest Method" can- 
not be advantageously applied, the interest being 
found without any calculation by simply removing 
the decimal point of the principal two places to the 
left. 



Per Cent 


. Days. Per Cent. 


Days. 


Per Cent. 


Days. 


Per Cent. 


Days. 


1 
t 


72O 


3i 


IO3* 


H 


55* 


9* 


38* 


I 


360 


4 


90 


7 


5 1 * 


10 


36 


*i 


24O 


4* 


80 


7i 


48 


10J 


34* 


2 


l8o 


5 


72 


8 


45 


II 


33* 


H 


144 


5* 


65* 


H 


42* 


ii* 


31* 


3 


I20 

ximately. 


6 


60 


9 


40 


12 


30 


*Appro: 





IJIJEI^T. 



TO COMPUTE THE INTEREST. 

PRINCIPAL, RATE ANDTIME BEINGGIVEN. 

THE TIME EXPRESSED IN DAYS. 

THE YEAR FIGURED AT 360 DAYS. 



100 DAYS INTEREST METHOD. 

Any principal showing its own interest at 36 per 
cent, for 1000 days, the interest for 100 days, 10 
days or 1 day can be readily seen by simply removing 
the decimal point i, 2 or 3 places to the left. 

For instance, the interest on 
3371.43 at 36 per cent, for 1000 days is $371.43 



00 


(( 


37-14 


10 


a 


3-7i 


I 


a 


•37 



—10— 

Taking the interest at 36 per cent, for 100 days, 
which cover the ordinary business terms, as basis, 
the same divided by 

3 gives the interest for 100 days at 12 per cent. 

4 " " " " 9 " 

g (( a k u £ a 

8 " " " " 4| " 

9 4 

and by pointing off 1 or 2 places, the interest for 
10 days or one day at the above rates. 

The divisors for the most frequent rates can be remembered 
without difficulty, being the result of 36 divided by the given 
rate. 

From these fundamental rates the interest at other 
rates can be conveniently obtained by adding or 
deducting the proper aliquot parts, for instance : 
6|- per cent, and 5% per cent, equal 6 per cent. 

more or less one -twelfth. 
7 per cent, and 5 per cent, equal 6 per cent, 
more or less one-sixth. 
Etc., etc. 

To compute the interest at 8 per cent., find first the interest 
at 4 per cent, and double the same,which is shorter than com- 
puting the interest at 6 per cent, and adding one-third. 

The most simple and practical methods of finding 
the interest at any given rate for 100 days, 10 days 
or 1 day, by pointing off 1, 2 or 3 places from the 
principal, are presented in condensed form in the 
following 



-u- 



TABLE. 



RATES. 


DIVISORS. 


PARTS. 


Per cent. 


Principal 
divided by 


To be deducted. 


i 


36 




2 


• 18 




3 


12 




3i 


9 


One-eighth. 


4 


9 




4 


8 




5 


7 


One-thirty-sixth. 


5i 


6 


One-twelfth. 


6 


6 




6i 


5 


One-tenth. * 


7 


5 


One-thirty-sixth. 


7i 


4 


One-sixth. 


8 


4 


One-ninth. 


8* 


4 


One-eighteenth. 


9 


4 




IO 


3 


One- sixth. 


ii 


3 


One-twelfth. 


12 


3 





* Approximately. 

NOTE. 
To deduct one-thirty-sixth deduct one-sixth of one-sixth. 
" one-eighteenth " one-third of one-sixth. 
" one-twelfth " one-half of one-sixth. 

The division of the principal is carried out to the 
tens of the cents. This is a new departure and in- 
sures greater accuracy than the old usage of "give 
and take." Two. places of the result are pointed off 
as cents, the remaining figures represent dollars of 
interest for 100 days. The units of the cents of 
the principal can always be omitted, their interest 
being, as a rule, too small to cause any variation in 
the answer. This interest method, considering the 



-12- 



tens of the cents of the principal, gives more accu- 
rate results than other short methods, which either 
omit the cents entirely or correct the number of 
dollars. 

To arrive at the interest for any desired number 
of days, multiply the hundreds, tens or units of days 
by the corresponding amount of interest and add 
these products together. 

If the number of days is between 90 and 100, deduct from 
the interest for 100 days as many times the interest for 1 day 
as there are days less. The interest for 50 days is evidently one- 
half of the interest for 100 days, the interest for 25 days one- 
quarter of the interest for 100 days. The interest for 9 days 
equals the interest for 10 days less the interest for one day, 
which should be remembered whenever the number of days 
contains a 9 in the unit place. 

Examples. 
1. 

§723.19 at 6 per cent, for 93 days? 



6)723-1 



12*05 
12X7= '84 



9 

= Interest at 6 % for 100 days. 

Answer: $11*21 =Interest at 6 % for 93 days. 



$147.51 at 7 per cent, for 117 days? 

5)_MT5 1 
— 36? 6X6) 2*95 =Interestat 7] % for 100 days. 
49 81— " 1% " 100 " 



2-Sj =Interest at 7 % for 100 days. 

0*287X1= -29 = " " " 10 " 

0*028X7 = ' 2 ° = " _J1_ " 7 " 

Answer: $3*36 =Interest at 7 % for 117 days. 



—13— 

3- 

S56.87 at 8 per cent, for 55 days? 

9)56.8 j7 

X2 *63 ! =Interest at 4 % for 100 days. 
i- 2 6|= " 8 "_ 100 " 

0*126X5 '63 =Interest at 8 % for 50 days. 
0*012X5 -06 = " " " 5 " 

Answer : $ -69 =Interest at 8 % for 55 days. 

THE YEAR FIGURED AT 365 DAYS. 

100 DAYS INTEREST METHOD. 

Deduct from the principal its seventy-third part — 
in leap years its sixty-first — and apply the "ioo Days 
Interest Method," explained and illustrated in the 
preceding pages, or use the "ioo Days Interest 
Method" first and deduct from the interest its 
seventy-third part. 

NOTE. 
Allowing for everv Sio. of interest, 14 cents. 
5- " 7 " 

I. "I " 

the one-seventy-third part is approximately obtained. 

Example : 

$341.20 at 4^ per cent, for 49 days? 

8 )34i'2 

4-26 =Interest at 4-^ % for 100 days. 



i 2-13= « " " 
— 4 == « « " 


50 " 
1 " 


2*09 ^Interest at 4^ % for 
— 73 3 
Answer : $2*06 


49 days. 



—14— 

Any principal divided by 73 gives the interest for 
100 days at 5 per cent. Any principal divided by 
5 gives the interest for 100 days, 10 days or 1 day, 
at 7*3 per cent, by simply removing tbe decimal 
point 1, 2, or 3 places to the left. 

THE TIME EXPRESSED IN MONTHS AND DAYS. 

Multiply the number of months by 3, annex one 
cipher and add the given number of days. If the 
time is expressed as an interval between two dates, 
make use of the "Improved Time Calculator". The 
"100 Days Interest Method" is applicable in both 
cases. 

THE TIME EXPRESSED IN MONTHS. 

If the reduction to days cannot be conveniently 

accomplished, multiply one-fourth of the principal 

by one-third of the product of the rate by the number 

of months and remove the decimal point two places 

to the left. 

Example : 

S428. — at 6 per cent, for 7 months? 

4 )428 

107X14 6X7=42 

428 i=i 4 



Answer : 514-98 



THE TIME EXPRESSED IN YEARS, MONTHS 
AND DAYS. 

Compute the interest for the given number of 



—15— 

years first and find the interest for months and days 
according to the directions given under the corres- 
ponding title. 

THE TIME EXPRESSED IN YEARS. 

Multiply one-fifth of the principal by one-half the 
product of the rate by the number of years, and re- 
move the decimal point one place to the left. 
Example : 

Si 2 25. — at 5 per cent, for 4 years? 
Answer: S 245- — 5X4=20 

i=io 
Note. — In this instance the multiplication by 10 counterbal- 
ances the moving of the decimal point. 

TO FIND THE PRINCIPAL. 

INTEREST, RATE AND TIME BEING GIVEN. 

THE TIME EXPRESSED IN DAYS. 

THE YEAR FIGURED AT 360 DAYS. 

Multiply the interest by 4, divide by the product of 
one-ninth the number of days by the rate, and re- 
move the decimal point 3 places to the right. 

THE YEAR FIGURED AT 365 DAYS. 

Apply the rule stated above and add to the an- 
swer its seventy-third part. 

THE TIME EXPRESSED IN MONTHS. 

Multiply the interest by 3, divide by the product 
of one-fourth the number of months by the rate, 
and remove the decimal point 2 places to the right. 



-16- 



THE TIME EXPRESSED IN YEARS. 

Multiply the interest by 5, divide by the product 
of one-half the number of years by the rate, and 
remove the decimal point 1 place to the right. 

AMOUNT, RATE AND TIME BEING GIVEN. 
THE TIME EXPRESSED IN DAYS. 

Divide the amount by the by 1000 increased product 
of one-quarter the rate by one-ninth of the time, 
and remove the decimal point 3 places to the right. 

THE TIME EXPRESSED IN MONTHS. 

Divide the amount by the by 100 increased pro- 
duct of one-third the rate by one-quarter the time, 
and remove the decimal point 2 places to the right. 

THE TIME EXPRESSED IN YEARS. 

Divide the amount by the by 100 increased pro- 
duct of one-half the rate by one-fifth the time, and 
remove the decimal point 1 place to the right. 

TO FIND THE RATE. 

INTEREST, PRINCIPAL AND TIME 

BEING GIVEN. 

Apply the rules given for finding the principal, 
substituting the known principal for the unknown 
rate. 

TO FIND THE TIME. 
INTEREST, PRINCIPAL AND RATE 

BEING GIVEN. 
THE TIME TO BE EXPRESSED IN DAYS. 

Multiply the interest by 4, divide by the product 
of one-ninth the principal by the rate and remove 
the decimal point 3 places to the right. 



—17— 
THE TIME TO BE EXPRESSED IN MONTHS. 

Multiply the interest by 3, divide by the product 
of one-quarter the principal by the rate and remove 
the decimal point 2 places to the right. 

THE TIME TO BE EXPRESSED IN YEARS. 

Multiply the interest by 5, divide by the product 
of one-half the principal by the rate and remove 
the decimal point 1 place to the right. 



DIJQOUflS. 

MERCANTILE DISCOUNT. 

SIMPLE DISCOUNT. 

Multiply the list price or amount of bill by the 
given rate and remove the decimal point 2 places to 
the left, or multiply the discount for 1 and 10 cents 
1, 10, 100, etc., dollars by the corresponding figures 
of the amount from which the discount should be 
deducted. 

Examples. 
1. 2. 

$173.21 less 5 per cent. $14.60 less 15 percent. 

$10. — Discount $1.50 

173.21 4. — " .60 

5 .60 " .09 



Discount : $8.66 $14.60, Discount : $2.19 



— IS— 

COMBINATION DISCOUNT. 

Subtract each of the given rates from ioo, multi- 
ply the differences and point off two places for every 
rate. The list price or amount of bill multiplied by 
this product gives the desired net price or net 
amount. Or consider the product of the rate differ- 
ences as the net of Si.oo expressed in cents and 
decimals, deduct the same from $1.00 to arrive at the 
discount and obtain the discount for any given 
amount by multiplying the discount for i and 10 
cents, i, 10, ioo, etc., dollars by the corresponding 
figures of the amount from which the discount 
should be deducted. 

Example : 
Amount : $425. — Discount, 40, 20 and 5 per cent. 

100 100 100 

— 40 — 20 — 5 1. 00 

60 X 80 X 95 =456000* 

•544 

S400. — Discount: S2 17.60 

25. " 13-60 

$425. — Discount: S231.20 

The computation of Trade Discounts can be con- 
siderably simplified by using the "Combination 
Discount Calculator." 



^Ciphers in the lowest places need not be considered. 



—19— 

BANK DISCOUNT. 
Any note giving its own interest at 36 per cent., 
if the year is figured at 360 days, or at 36^ percent, 
if the year is figured at 365 days, for 100 days, 10 
days or 1 day by removing the decimal point, 1, 2 or 
3 places to the left, the interest for any given num- 
ber of days at 36 or 36^ per cent, can be easily 
found by multiplying the interest on each note for 
1, 10 or 100 days by the units, tens or hundreds of 
the corresponding number of days. These products 
added together give the Interest or Bank Discount 
on all notes at 36 or 36-J per cent. 

To obtain the interest at any given rate, divide 
the sum of the products by one of the divisors of 
the table, p. 1 1, according to the rate and deduct the 
part which the corresponding column indicates, or 
compute first the interest at 6 per cent, by dividing 
by 6, and add or deduct the proper aliquot parts. 

NOTES. 

The dollars of any principal show the interest in cents for 
10 days at 36 or 36^ per cent.; ten times the interest for 10 
days equals the interest for 100 days, and one-tenth of it the 
interest for one day. 

In computing interest, the fractions of cents should be 
taken in consideration; for instance, the interest on S345.63 
at 36 or 36^ per cent, respectively is for 100 days, S34.56, 10 
days, 53.46, 1 day, 35 cents. 

The tens of the cents of the principal are considered when- 
ever the time exceeds 100 days, which secures the greatest 
accuracy for this method. 



-20- 



Example : Bank Discount on following notes, 7% ? 



. 09 T. 

: 28 



2.1 8" 
9.3 6 " 
7.5 o " 
0.2 9 " 



for 16 days f 3.40 

2.04 



29 

34 

80 

112 



f 1-44 

1 .65 

\ 3.88 

1 .52 

7.00 

U.03 

\ 40 

I 8 

5 ) 2 344 

4.69 

— one-thirty-sixth, . 1 3 

Answer : $4.56 
From the above answer one-seventy-third part 
must be deducted, if the year is figureo>at 365 days. 

TRUE DISCOUNT. 
To ascertain the amount which, after de- 
ducting the Interest or Bank Discount at a 
given rate and for a given number of days, 
leaves a given principal or present value. 

Multiply one-sixth of the rate by one-sixth of the 
number of days, remove the decimal point three 
places to the left, subtract the product from 1 unit, 
and divide the given principal by this difference. 

Example : 

Present Value, $1000. — , 6 percent. 95 days. 
Amount ? 
6)6__ 6 )93 

1 X 1 — o-oi5-5=o'9845) iooo.oo($ioi5*74, Ans. 



—21— 

To ascertain the principal or present value 
which at a given rate and in a given number 
of days, produces a given amount. 

Multiply one-sixth of the rate by one-sixth the 
number of days, remove the decimal point three 
places to the left, add one unit to the product and 
divide the given amount by this sum. 

Example : 

Amount, Si 500. — , 9 per cent. 48 days. 
Present Value ? 
6)96)48 

1*5X8, i-|-o , oi2 = i , oi2) 1500.00(31482*21, Ans. 

NOTE. The most convenient rate for these calculations is 
6 per cent., as the division by 6 reduces it to 1 and saves its 
consideration. 

The amount or principal can also be found by di- 
viding the given number of days by one of the di- 
visors of the Table, page 11, according to the rate, 
deducting the corresponding part, removing the 
decimal point 3 places to the left, subtracting or 
adding the difference from or to 1 unit, and dividing 
the given principal or amount by this difference or 
sum. 

The "True Discount" is obtained by deducting 
the present value from the amount. 



TO FIND THE COMPOUND INTEREST. 
PRINCIPAL, RATE ANDTIME BEING GIVEN. 

The compound interest of Si. 00 can be obtained 
by multiplying successively the rate, squared rate, 



—22— 

cubed rate, etc., by the factors of the following 
table, according to the number of years, placing 
each following product 2 places to the right under 
the one preceding, and finding their total. 

COMPOUND INTEREST FACTORS. 









b 




8 




s 


1— 1 



8 

O 
O 
O 


1 

3 




8 


1 







03 

CD 


<d 
4-3 

c8 


X 

02 


O 

X 



p 

© 



8 

O 


8 


8 

O 
O 




8 


&H 


Ph 


4-> 

c3 




X 


O 


O 
O 


8 


8 


O 

5 







M 


+3 


1 


X 


O 


O 
O 


8 


5 
O 


1 

2 


1 

2 


1 




CD 


ua 

CD 


X 


O 

X 


b 


O 
O 
O 
O 











r^ 


43 






X 


b 


3 


3 


3 


1 






CD 


1 


1 


X 


4 


4 


6 


4 


1 






CD 



















Ph 


*J 


• 


1 





5 


10 


10 


5 


1 




« 


3C 

CD 


1 
















+3 




6 


6 


15 


20 


15 


6 


1 








CD 
43 


7 


7 


21 


35 


35 


21 


7 


1 






8 


8 


28 


5(3 


70 


56 


28 


8 


1 




9 


9 


36 


84 


12(3 


126 


84 


36 


9 


1 


10 


10 


45 


120 


210 


252 


210 


120 


45 


10 



8 



1 



Note. — These factors, being the members of progressions 
which stand in simple relations to each other, can be remem- 
bered without difficulty and are readily extended to any given 
number of years. For practical purposes it is sufficient to pro- 
ceed only to the 3d power of the rate. 

Example : 
Compound Interest on $ 1 .00 for 4 years at 5 per cent ? 

5X4 =2 °- • 
25X6= 150. . 

125X4= 5°°- • 
625X1= 625 

$0*21550625 = Answer. 



—23— 

To ascertain the compound interest on any given 
principal, multiply the same by the compound in- 
terest of 3 1. oo. 

To find, approximately, the number of years in 
which any principal doubles itself through the ac- 
cumulated compound interest, divide 72 by the given 
rate. 

Compound interest, if computed semi-annually or 
quarterly, is equivalent to the annual compound in- 
terest for twice or four times the number of years 
at one-half or one-quarter the rate. 

TO FIND THE PRINCIPAL. 

COMPOUND INTEREST, RATE AND TIME 
BEING GIVEN. 

Divide the given compound interest by the com- 
pound interest of Si. 00 at the given rate and for 
the given number of years. 

TO FIND THE RATE. 

COMPOUND INTEREST, PRINCIPAL AND 
TIME BEING GIVEN. 

Ascertain the compound interest of Si. 00 by di- 
viding the given compound interest by the 
given principal, increase the same by 1 and extract 
the root which the number of years indicates. De- 
duct 1 from the result, the difference representing the 
desired rate. Whenever the number of years is 
such that square and cube roots cannot be applied, 
the use of logarithms is indispensable. 



LIBRARY OF CONGRESS 



-24— 



TO FIND Th 021 060 446 7 
COMPOUND INTEREST, PRINCIPAL ANU 
RATE BEING GIVEN. 

Ascertain the compound interest of $i.oo by di- 
viding the given compound interest by the given 
principal — increase the same by i and divide this 
sum successively by ioo plus the given rate, until 
the quotient equals unity. The number of divisions 
indicates the time in years. 

ANNUITIES. 

To find the amount to which an annual investment 
of $ i. oo will accumulate, divide the compound in- 
terest at the given rate and for the given number of 
years by the rate, and remove the decimal point two 
places to the right. 

Example : 
Annual Investment, $i.oo. 5 per cent. 4 years. 
Accumulated Amount ? 
5)0*2155 ( £4-31, Answer. 
To find the annuity with which to pay $1.00 in a 
given number of years, divide the rate by the com- 
pound interest, add the rate to the quotient, and 
remove the decimal point two places to the left. 

Example : 
Debt, $1.00. 5 per cent. 4 years. Annuity? 
0*2155) 5*00000 (23-24-5=5 — .28-2, Answer. 
The result obtained for $1.00 can be applied to 
any amount by multiplying the former by the latter. 



